Does there exist a stochastic process $X$ such that when two such trajectories are sampled, their composition is Wiener-distributed? It would be natural to call such a process a functional square root of Wiener process, and take a look at some sampled trajectories.
A related question is: given a typical Wiener trajectory $f \colon \mathbb R \to \mathbb R$, does there exist $g \colon \mathbb R \to \mathbb R$ such that $g \circ g = f$? It seems that a Wiener process is in some sense "compressing" the real line ($W(t) \approx \sqrt t$) so it is "very surjective" suggesting that the answer may be yes, but I don't know how to prove it. Also, it seems that the asymptotics of such functions may be $t^{1/\sqrt2}$, as $(t^{1/\sqrt2})^{1/\sqrt2} = \sqrt t$.
P.S. Thanks to a comment by Ilya, indeed, I forgot to mention that I mean the two-sided Wiener process, so that a trajectory is a function $\mathbb R \to \mathbb R$, not $\mathbb R_{\geq0} \to \mathbb R$. This construction is also intuitive and translation-invariant, for example.
